Journal of Statistical Physics

, Volume 166, Issue 3–4, pp 926–1015 | Cite as

Entropic Fluctuations in Thermally Driven Harmonic Networks



We consider a general network of harmonic oscillators driven out of thermal equilibrium by coupling to several heat reservoirs at different temperatures. The action of the reservoirs is implemented by Langevin forces. Assuming the existence and uniqueness of the steady state of the resulting process, we construct a canonical entropy production functional \(S^t\) which satisfies the Gallavotti–Cohen fluctuation theorem. More precisely, we prove that there exists \(\kappa _c>\frac{1}{2}\) such that the cumulant generating function of \(S^t\) has a large-time limit \(e(\alpha )\) which is finite on a closed interval \([\frac{1}{2}-\kappa _c,\frac{1}{2}+\kappa _c]\), infinite on its complement and satisfies the Gallavotti–Cohen symmetry \(e(1-\alpha )=e(\alpha )\) for all \(\alpha \in {\mathbb {R}}\). Moreover, we show that \(e(\alpha )\) is essentially smooth, i.e., that \(e'(\alpha )\rightarrow \mp \infty \) as \(\alpha \rightarrow \tfrac{1}{2}\mp \kappa _c\). It follows from the Gärtner–Ellis theorem that \(S^t\) satisfies a global large deviation principle with a rate function I(s) obeying the Gallavotti–Cohen fluctuation relation \(I(-s)-I(s)=s\) for all \(s\in {\mathbb {R}}\). We also consider perturbations of \(S^t\) by quadratic boundary terms and prove that they satisfy extended fluctuation relations, i.e., a global large deviation principle with a rate function that typically differs from I(s) outside a finite interval. This applies to various physically relevant functionals and, in particular, to the heat dissipation rate of the network. Our approach relies on the properties of the maximal solution of a one-parameter family of algebraic matrix Riccati equations. It turns out that the limiting cumulant generating functions of \(S^t\) and its perturbations can be computed in terms of spectral data of a Hamiltonian matrix depending on the harmonic potential of the network and the parameters of the Langevin reservoirs. This approach is well adapted to both analytical and numerical investigations.


Harmonic networks Fluctuation relations Large deviations Entropic functionals 



This research was supported by the CNRS collaboration grant RESSPDE. The authors gratefully acknowledge the support of NSERC and ANR (Grants 09- BLAN-0098 and ANR 2011 BS01 015 01). The work of C.-A.P. has been carried out in the framework of the Labex Archimède (ANR-11-LABX-0033) and of the A*MIDEX Project (ANR-11-IDEX-0001-02), funded by the “Investissements d’Avenir” French Government programme managed by the French National Research Agency (ANR). The research of AS was carried out within the MME-DII Center of Excellence and supported by the RSF Grant 14-49-00079.


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© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsMcGill UniversityMontrealCanada
  2. 2.Aix-Marseille Université, CNRS, CPT, UMR 7332, Case 907MarseilleFrance
  3. 3.Université de Toulon, CNRS, CPT, UMR 7332La GardeFrance
  4. 4.FRUMAMMarseilleFrance
  5. 5.Department of MathematicsUniversity of Cergy-Pontoise, CNRS, UMR 8088Cergy-PontoiseFrance

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